What comes next?
example for a hardness result:
cross×plain→cross, ’all operations’ is Max SNP-hard
wr +wb
2 ).
S.Will, 18.417, Fall 2011
(i.e. without the restriction wa =
Max-Cut-3
max cut
• formal:
• let G = (V , E ) be a graph
• a cut in G is a set of edges s.t. there is a partition
v2
v3
v1
v6
V1 ] V2 = V , where for every edge one endpoint is
in V1 , the other in V2 .
• Max-Cut-3: given graph g with degree ≤ 3, find cut with
maximal cardinality.
v4
v5
Theorem
Remark An optimization problem is Max-SNP-hard iff it does not
have a PTAS (Polynomial Time Approximation Scheme).
A PTAS is an algorithm that takes an instance of a maximization
problem and a parameter > 0 and, in polynomial time, produces
a solution that is within a factor 1 − of being maximal.
S.Will, 18.417, Fall 2011
Max-Cut-3 is Max-SNP-hard
Reduction of Max-Cut-3 to cross×plain→cross
Reduction idea:
represent Max-Cut-3 problem as alignment problem
cross×plain→cross such that optimal alignment corresponds
to maximum cut.
→ if Max-Cut-3 can be solved using the alignment problem, the
alignment problem must also be Max-SNP-hard.
Plan
• show how to represent graph G as input of alignment problem
• show how optimal alignment corresponds to maximum cut for
G.
S.Will, 18.417, Fall 2011
(e.g. Sequences S1 ,S2 + structure P1 for S1 )
Representation of Graph G as Alignment Problem
(Example)
v2
v3
v1
v4
AAAUUU
AAAUUU
AAAUUU
AAAUUU
AAAUUU
AAAUUU
UUUAAA
UUUAAA
UUUAAA
UUUAAA
UUUAAA
UUUAAA
v1
v2
v3
v4
v5
v6
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v5
v6
Representation of Graph G as Alignment Problem
(formally)
v2
v3
• given G = v
1
v4
v6
• sequences
v5
• S1 = (AAAUUU(C )c )n−1 AAAUUU, and
• S2 = (UUUAAA(C )c )n−1 UUUAAA.
nodes
• each edge (vi , vj ) of G corresponds to two arcs in P1 : one connecting
an A of the i-th segment with a U of the j-th segment and one
connecting a U of the i-th segment with an A of the j-th segment.
• C s are used to avoid alignment of different segments, and their
← arc changes
a ,wr )
number c depends on the ratio min(wwb ,w
← base deletion
d
S.Will, 18.417, Fall 2011
• the segments AAAUUU in S1 and UUUAAA in S2 correspond to the
Correspondence of Optimal Alignment and Max Cut
Properties of Optimal Alignment
• we choose c such that every optimal alignment must match all C s
• we choose a scoring with wm > wd and 2wa > wb + wr .
AAA UUU
• wm > wd implies no base mismatch: A A A U U U >
UUUAAA
• two alignment types for each node vi :
• A-type:
• U-type:
UUUAAA
AAA UUU
UUUAAA
AAA UUU
UUUAAA
• A-type :⇔ node in V1
U-type :⇔ node in V2 .
• cost for each edge of the cut (vi and vj have different type)
AAA UUU
UUUAAA
cost: wb + wr
AAA UUU
UUUAAA
S.Will, 18.417, Fall 2011
arc breaking
arc removing
Correspondence of Optimal Alignment and Max Cut
• cost for each edge that is not in the cut (vi and vj have same type)
arc altering
arc altering
AAA UUU
AAA UUU
UUUAAA
UUUAAA
•
•
•
•
C = k (wb + wr ) + (
V1 = all A-type nodes
V2 = all U-type nodes
n nodes, each degree 3 ⇒
k := |cut(V1 , V2 )|
3n
2
edges
3n
− k) 2wa + n 3wd
2
assumption: 2wa > wb + wr
⇒
z
}|
{
= 3n(wa + wd ) − k (2wa − wb − wr )
• ⇒ C minimal ≡ k maximal
• ⇒ maximal cut ≡ minimal edit distance.
>0
S.Will, 18.417, Fall 2011
cost: 2wa
• total cost for alignment:
Approaches for Alignments of RNAs
Plan A
Plan C
Plan B
A:
B:
ALIGN
FOLD
single
sequences
single
sequences
A:
A:
B:
FOLD
alignment
simultanously
ALIGN and FOLD
B:
[Sankoff 85]
ALIGN
sequence AND
structure
A:
B:
consensus:
consensus structure:
A:
B:
adopted from:
[Gardener & Giiegerich BMC 2004]
S.Will, 18.417, Fall 2011
consensus
structure
Simultaneous Alignment and Folding: Sankoff (1985)
• What do we want? What means folding into a common structure?
• First idea: preserve “shape” ≡ branching structure
• Formally: let i1 < i2 . . . < iv in a and j1 < j2 . . . < jw in b be the
• Hence: minimize edit distance + energies (of 2 equiv. structures)
S.Will, 18.417, Fall 2011
positions in pairs that limit multiloops or are external
(branching configuration)
Then: structures equivalent (according to branching)
iff v = w , and (if , ig ) ∈ Pa if and only if (jf , jg ) ∈ Pb
• finding good equivalent structures not sufficient:
Sankoff Problem Definition
• Idea: Sankoff = Zuker Folding + Needleman/Wunsch Alignment
• IN: two sequences a and b
• find two equivalent structures Pa and Pb
and compatible alignment A of a and b
such that Energy (a, Pa ) + Energy (b, Pb ) + EditDistance(A) minimal
• where: Energy yields (loop-based) Turner free energy,
EditDistance is edit distance (base mismatch x, indel y)
alignment must be “consistent” with branching structure
formally: the base pairs (if , ig ) ∈ Pa and (jf , jg ) ∈ Pb (from Def.
of equivalent) must be aligned to each other
S.Will, 18.417, Fall 2011
• what means compatible?
Constraints
We want to find the optimal structures + alignment with the
following constraints:
constraints on the predicted structures:
• must be equivalent (intuitively: same kind of multiloops)
constraints on the alignment:
• multiloops must be aligned to their equivalent partner
• hairpin loops must be aligned to their equivalent partner
one other 2-loop or must be entirely aligned to a gap.
S.Will, 18.417, Fall 2011
• each 2-loop (or stacking or bulge) must be aligned to exactly
Edit distance of sub-sequences
• distance based score
• Recursion:
D(i, j − 1; h, k − 1) + x



D(i, j − 1; h, k − 1)
D(i, j; h, k) = min

D(i, j − 1; h, k) + y



D(i, j; h, k − 1) + y

D(i + 1, j; h + 1, k) + x



D(i + 1, j; h + 1, k)
= min

D(i + 1, j; h, k) + y



D(i, j; h + 1, k) + y
(
x
• Initialization: D(i, i; h, h) =
0
if ai 6= bh
else
if aj 6= bk
if aj = bk
if ai 6= bh
if ai = bh
S.Will, 18.417, Fall 2011
x = base mismatch
y = base deletion/insertion
• D(i, j; h, k) minimum sequence alignment cost
between sequences ai . . . a
j and bh . . . bk .
Recall Zuker
Energies: e(s), where s is k-loop (or s = φ for empty structure)
F (i, j) “free”, minimum energy for subsequence ai . . . aj
C (i, j) “closed”, minimum energy for subsequence where (i, j) ∈ P
Zuker Recursion:
• Problem: (6) requires time proportional to n2K
where K maximum k in k-loops
S.Will, 18.417, Fall 2011
•
•
•
•
Usual Simplification
• e(s) for k-loops with k ≥ 3 (multiloops)
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e(s) = A + (k − 1)P + uQ
• New matrix: G (i, j) for multiloops
• Recursion:
Simultanous Alignment and Folding
• Extend definition of D(i1 , j1 ; i2 , j2 )
S.Will, 18.417, Fall 2011
if i1 > j1 , then cost for deleting bi2 . . . bj2 .
if j2 > i2 , then cost for deleting ai1 . . . aj1 .
• F (i1 , j1 ; i2 , j2 ) minimum cost (sum of alignment and free energy)
for ai1 . . . aj1 and bi2 . . . bj2 .
• C (i1 , j1 ; i2 , j2 ): minimum cost for ai1 +1 . . . aj1 −1 and bi2 +1 . . . bj2 −1
under condition (i1 , j1 ) ∈ Pa and (i2 , j2 ) ∈ Pb
S.Will, 18.417, Fall 2011
Simultanous Alignment and Folding: “Closed”
• G (i1 , j1 ; i2 , j2 ): matrix for multiloop alignment
• Recursion for G
G (i1 , j1 ; i2 , j2 )

match j1 and j2
match i1 and i2



}|
{
}|
{
z
z



C (i1 , j1 ; i2 , j2) + 2P + D(i1 , i1 ; i2 , i2 ) + D(j1 , j1 ; j2 , j2 )





G (i1 , h1 ; i2 , h2 ) + (j1 − h1 + j2 − h2 )Q





= min

+D(h1 + 1, j1 ; h2 + 1, j2 ),




min
G (i1 , h1 ; i2 , h2 ) + G (h1 + 1, j1 ; h2 + 1, j2 ),



i1 < h1 < j1 





 i2 < h2 < j2 
(h1 − i1 + 1 + h2 − i2 + 1)Q




+D(i1 , h1 ; i2 , h2 ) + G (h1 + 1, j1 ; h2 + 1, j2 )
S.Will, 18.417, Fall 2011
Simultanous Alignment and Folding: Multiloop
Simultanous Alignment and Folding: “free”
• Recursion for F


C (i1 , j1 ; i2 , j2 ) + D(i1 , i1 ; i2 , i2 ) + D(j1 , j1 ; j2 , j2 )


min
F (i1 , h1 ; i2 , h2 ) + F (h1 + 1, j1 ; h2 + 1, j2 )
F (i1 , j1 ; i2 , j2 ) = min
i1 < h1 < j1
 i2 < h2 < j2



D(i1 , j1 ; i2 , j2 )
• with initial conditions C (i1 , i1 ; i2 , i2 ) = ∞
S.Will, 18.417, Fall 2011
and G (i1 , i1 ; i2 , j2 ) = G (i1 , j1 ; i2 , i2 ) = ∞
Complexity
space complexity O(n4 )
• constant number of matrices (C,D,F, and G)
• each of them has O(n4 ) entries
time complexity O(n6 )
• each entry of matrix D requires constant time
• each entry of F,C, and G requires O(n2 ) time (minimize over
• hence: n4 · n2 = n6
S.Will, 18.417, Fall 2011
all h1 , h2 )